Limestone around pine trees? Bunch of mushrooms?

The neighbor has 4 very large pine trees that adjoin my front yard, underneath them and as far out as the needles can fall my grass growth is poor... if we give the rest of my yard a 8 out of 10 this area is more like a 5.

I have a bag of limestone and a broadcast spreader, should I just spread in that area and if so... what setting on the dial?

I rake the needles up twice a year and water/fertilize this area like anywhere else in the lawn.

Also, since I aerated and used starter fertilizer I have gotten a few dozen big mushrooms all over every week or so, a buddy down the street is getting the same i've just been mowing over them because they're rooted good and spill junk all over if you try to pull em out... is it a problem?

I'll attach a pic showing what condition the lawn's in and these mushrooms, the darker areas are where crabgrass tried to take over this year... even this late the grass is still growing (cold here in west michigan already) and recovering.